Effect of disturbances on beating and decay: Zeno paradox without observations

Volume 134, number 7

PHYSICS LETTERS A

23 January 1989

EFFECT OF DISTURBANCES ON BEATING AND DECAY: ZENO PARADOX WITHOUT OBSERVATIONS P.T. GREENLAND and A.M. LANE Theoretical Physics Division, AERE, Harwell, Oxon. OXIJ ORA, UK Received 29 June 1988; revised manuscript received 14 November 1988; accepted for publication 22 November 1988 Communicated by B. Fricke

We consider the effect of disturbances on decay transitions and beating transitions. The disturbances may be internal to the system, or external (collisions). We expose the common elements in a wide range of situations, stressing the inhibition of a transition that results from sufficiently strong or frequent disturbances. In the limit, transitions are completely suppressed, thereby giving a parallel to the quantum Zeno paradox, but with the difference that no observations are involved.

1. Introduction

(unobserved) amount 212 by a factor TI This effect of repeated observation in changing the coherent quadratic law into the incoherent linear law is the essence of the quantum Zeno paradox, which states that the transition probability becomes zero if T-+0, i.e. observations are infinitely frequent (continuous). If, instead of observation, the state is disturbed at 1= T 1, in the sense of the introduction of a relative I.

Many situations revealing the effects of a disturbance on beating or decay have been mentioned in the literature. As far as we know there has been no previous attempt to unify and systematise these phenomena. Our object is to do this. Not only are they closely related among themselves, they also have close conceptual contact with the effect of repeated observation on beating or decay, the so-called “quanturn Zeno paradox”, which has received much attention in the last ten years, recently, e.g. in refs. [1—3]. The essential common ground is readily described. We are given a bound state 10> at time 1=0, the solution to a hamiltonian H0. Under an extra interaction, H’, switched on at I = 0, there will be a coherent transition to another state of H0, I n> say, so the total wave-function is, for small time t and settingh=l:

I ~P(t)>= 10> —tIn> 1

(1)

and probability at = Tthe is corresponding 2T2.transition If an observation oftime the “transitted” part is made at 1= T, then the surviving part 10> starts its transition anew at this time T. If observations are repeated up to a time 1>> T, then the accummulated transition probability is H’ 10> 2T2(lIT), which is less than the free

phase-change a1 between 10> and I n> then, for 1> T1,

W(t)=I0>—iIn>[e’~’T1 +(t—T1)J (2) After many such disturbances i at intervals T,, the accumulated transition probability is 2

e~


10>2.

This is less than the free probability if the a are not (lIT). such all equal. If a~are random2and largeThus (mean I a•Jphase~ then the first factor is have T just the same effect as changing disturbances observations in inhibiting the transition, so we may speak ofa “Zeno paradox without observations”. We note that, in the case of more than one state I n>, these results are unchanged, the only affect being to introduce a sum on n, which will become an integral 429

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if the states I n> form a decay continuum. Of course, in the case of decay, the undisturbed system eventually generates its own incoherence, beginning at a time equal to the inverse of the energy band width around 10> of significant . The range of coherent states I n> falls as time increases until all states, except these degenerate with 0>, are lost. (See e.g. ref. [41.)) This happens at a time of order of the inverse of the kinetic energy release. Thus the decay becomes linear with “golden rule” width, essentially the energy density of 2 at state 10>. Turning now to descriptions of disturbances and their effects, we focus on the two most common timedependent phenomena, quantum beats between two states, and the decay of a prepared state. We consider the effect of “internal” and “external” disturbances, arises fromrepresenting a time-independent additionthe to former the hamiltonian, coupling to “spectator” degrees of freedom, while the latter involves an explicitly time-fluctuating addition. A wide range of physical phenomena are covered by these situations, and we mention them in the text below. In the literature, these have been discussed separately, using a wide range oftheoretical methods, and this distracts attention from their considerable cornmon ground, viz, for rapid disturbances, whether internal or external, the coherent time development implied by the original hamiltonian is inhibited, and tends to be “frozen” at a time of order of the inverse of the disturbance frequency. We will see this effect appearing in all of the situations that we now discuss.

2. Beating with collisions

—1~oo=

P11

Poo

where

=

sec /iet/4Tc0s (pt /3) —

—

,

p2=d2—(4T)-2, cot/l=4Tp.

(4) (5)

We see that, as T_

1 increases, the beating frequency falls steadily from d at T = 0 to zero at T —1 = 4d. Furthermore the beating is damped at rate (4T) For T ~> 4d, the solution is purely decaying: —‘

—

Pu

Poo =

—

{(l

—y) exp[(

—

t/4T)(l +y)]

—(y+l) exp[(—t/4T)(l—y)]} (6) 2]1”2. For T~~.4d, y~1 and where y=_ [1— (4Td) the second (slow) exponential dominates. The characteristic transition time from I 0> to I 1> is (2d2 T) which ~ as T Thus, for sufficiently rapid disturbances, transitions are eliminated. This is our first example of the Zeno effect without observations. In passing, we note that a recently suggested model for quantum observation (rather than excitation) of a two-state oscillation [7] leads to equations of type (3)). A practical application of these ideas (besides the case of Rabi oscillations), is muonium—anti-muonium beating. The muon lifetime z is the expected beating time d’, so the transition probability per muonium is ~dt in the absence of collisions. Collisions reduce this by a factor 2T/r (from (6) using T>> t>> d’). For a gas at NTP, this is estimated as ,

~,

—~

- ‘—~ ~.

‘~

3. Beating disturbed by internal coupling The simplest model for this is where the disturbance arises from the coupling of the beating states to a spectator degree of freedom represented by two

~id(p 10—Poi)

l~uo=—~-~p10—~id(p00—p11) ,

states, viz, the “vacuum state”, v, and “one phonon” state, k: (3)

where d is the Rabi frequency, assumed equal to the energy difference between levels 0, 1. Starting with 430

small T~(<4d), the solution for initial condition Poo = 1 is readily shown to have the form:

l0~ [8], showing the strong inhibiting effect of collisions.

This situation is met in several contexts, perhaps the best known being Rabi oscillations in which two levels of a system (0,. 1 say) are coupled by a laser field. In the presence of phase-changing collisions characterised by time interval T, the density matrix p satisfies (see e.g. refs. [5,6]): Pi i =

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PHYSICS LETTERSA

For small coupling a << 1 and w>> d the beating is slowed by a fractional amount linear in a, and con-

H=~d(I0><11 + 11> <01) + JCOk( 1k>
+Hk(IO>
—

—

Iv>
I l><1 I )(Iv>
,

(7) When a term does not involve a degree of freedom, the unit operator for that degree is assumed. In the absence ofbeating (d= 0), states 10>, Il> are shifted equally by the coupling, so any effect on beating is not a trivial energy-shift. For small H~< I d2 I, the effect of the coupling on the beat frequency is —

d

—~

d’ d( 1 =

H~ ~ —

—

d2)’

We see that beating is slowed if cok> d, but not if < d. For large Hk—’ d’ -~0 for d, but d’-.2Hk for cok
~,

23 January 1989

sistent with the above result summed on k; also the beating is damped at a rate ~,xda.For strong coupling, represented by a approaching the value the ~,

beating frequency and damping rate go to zero, as (~ a )2 ~ a respectively. In this sense, we have a Zeno effect, but note that its origin lies in the increasing finite coupling to many oscillators rather than infinite coupling to one osicalitor of type (7) or (8). Thus it has a lot in common with the case of external disturbances (collisions). Another model for internal disturbance allows state I 1> to decay. Rather than couple I 1> directly to the continuum, we do so indirectly via a state 12>: H= ~d( 10> <11 + I 1> <01) +A 12> <21 —

—

(Ok>

bances of beating is

J

+H’ (Il> <21 + 12> <11) + dEh~(12> <21)

The common energy of states 10>, Il> is set equal to zero. If hE is constant with energy (= h) and we ignore threshold effects (i.e. take the lower integration limit to E—+ ~), then the problem is readily solved. It can be shown that the effect ofthe last three terms on the (10>, Il>) system can be reproduced by their replacement by —

JdEIIE(Il>
(10)

where H’ 2h2 E

H=~d(0> <11 + Ii> <01)

(9)

.

(11)

(E—4)2+(tth2)2’

It is readily shown that the solution of (9),

+Hk(I0>
~P(t)=aoI0>+a 1I1>+a2e~’I2>

+wkak~ak,

(8)

where [a~, ak] = 1. This hamiltonian allows the excitation of more than one phonon in each mode. For weakabove coupling, onebut readily confirms agreement with the result, no analytic solution exists in general [9]. Ifthe last two terms in (6) are summed over k up to an upper limit, w~say, an approximate solution has been given, when H~is such that its energy density times co~is constant, a/2 say [10,11].

(12)

+JcIEaEe_~tIE>, is such that any one of the amplitudes 21, where a0, a1, sum of three exponentials, e A~+(iA+ xh2)~.~.2+ (H’2+~d2)A + ~ xh2d2 + ~id24 = 0.

...,

is a

(13)

For small H’2 the values are 431

Volume 134, number 7

—

+

(~+xh2)(l

~id

—

—

PHYSICS LETTERS A

—

)

“golden rule” decay, characterising complete loss of greater thanFor the time for onset with of thetime linearintervals region, coherence. disturbances

(~+xh2)2+ ~d2 , H’2

~h1~2

i (4 ±~d) + ith2’

so that, to first order the beating frequency is unshifted, while the beating decays at rate itlI~,which is one half the decay rate of state I 1>. For large H’2, one exponential in a 0 dominates with coefficient 1 and slow decay rate = 2it (dh/2H’ ) 2 which 0 as H’ ~ce~’ An example ofthis inhibition ofbeating caused by the decay ofone of the two states is that of nucleon— anti-nucleon beating when the nucleon is in a nucleus. The rapid decay of the anti-nucleon state (width of order 100 MeY) means that the free nucleon transition rate (order 10~22eV) is reduced by a factor l0~° [12,13]. ..~.

—

—~

4. Decay disturbed by (phase-changing) collisions This case has been recently discussed [14]. The essential result is that, for collisions at mean time interval T, the usual decay width r=2xh~,for a state I 1> at energy E 1 is replaced by the lorentzian average 2+T2’

TIJdE (EE1)h~ 0

(14)

Thus we see that, depending on the local shape of h F may, for small T be increased or decreased, but that since h~—+0as E—~0,~, as T’ increased, the decay width must ultimately be reduced and eventually to go zero. A related result is that, in this situation, the golden rule law of decay sets in at the time T if this is smaller than the usual onset time Ej* ~.,

~,

5. Decay disturbed by internal coupling The case of decay is somewhat more complicated than that of beating. As a decay transition develops in time, the initial “coherent” region of quadratic time dependence passes into a region of increasing incoherence before finally settling down to the linear 432

23 January 1989

the decay is unaffected. If the mean time interval between disturbances is now decreased below this time, then the decay may be speeded up or slowed, depending on dynamical details. The essential critenon is whether the golden rule width evaluated at resonance is larger or smaller than its value at other energies. If generally smaller, than the disturbed decay is likely to be speeded up for a range of mean time interval. Only when the interval is reduced to fall within the quadratic region will the decay be slowed, and ultimately eliminated. This can be seen explicitly with the model of eq. (9). We are now interested in how the decay of state Ii> is affected by increasing d, the coupling to another state, 10>, this representing disturbance 2< (taken to be zero). When d increases beyond 2xfi~,the decay rate falls to reflecting the beating of I 1> into the non-decaying state 10>. As d then increases towards A, there are two decaying components corresponding to the splitting of 10>, 1> into 2_1~/2( 10> ±Ii>) with energies ±d. The decay rates are ttfi~with E= ± d, so from (11), for d~ reduced by overlap factors of order (N+ 1) 1/2 (ii )The intermediate trend (for 2xfi~<
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PHYSICS LETTERS A

pendingonwhethertheshiftistowardsorawayfrom any peak in fi~[4]. (iii) The ultimate trend, for d>>4. This is a decrease to zero as d—~co,and occurs because the decaying states are ultimately shifted to beyond the bandwidth of l~.This is the Zeno effect. It was suggested [151 that proton decay of a proton in a nucleus was strongly reduced by scattering disturbances from other nucleons, and that this explained the lack of observed decays. It was subsequently pointed out that the time between disturbances (of order 1022 s) was much larger than the time needed to give a reduction (order 1025 s) and also larger than the time (order 10~23s) to have any effects [16,171. We note in passing that some comments (e.g. in ref. [18]) on the original suggestion maintained that there was an error of principle, viz, that transitions are not inhibited unless observational collapse occurs. Our model shows that this is incorrect (for reasons apparent in our introductor’ remarks). A decay may be affected by disturbances to its f~ nal (free) states IE> as well as to its initial state Ii>. A specific model hamiltonian that represents this situation is

23 January 1989

H=(El l>
+

J

dEEIE>
+ ~ H~L-~’ IaE>
/3>
.

a.fl

It is known that the width 1= 2xh~Eof a resonance state I la> is modified into r=P~’2r1~2, where ~l

/2 =

(2E)1 /2(1 —

(hE

x

+~

7LiKEE )

—

K~~ \ EE’

)~

(17)

and hE is the vector (haE, 0, 0, KEE is the background K-matrix generated by (16). In general, disturbance effects may be masked by two other effects: (a) the principal part term which may cause an increase or decrease; this is suppressed if we assume energy-independent H’, implying that K=H’ (= constant), (b) the effects of a change in the elastic scattering potential; this is associated with terms a=/i in (16), so we will drop these. With these assumptions one can see that F given by (17) is reduced below 2xh~~ by a factor that —~0as H’2 as ...).

E 1 Ii> <2 I

the typical magnitude of H’ increases. (For a separable potential H~=DaDp, explicit formulae are obtainable. One can see that this effect is lost ifterms a = /1 are retained.) From our general picture of the effect of disturbances, we expect that this reduction

+ ~H’ (I/J> <21 +

+

$ $

dEEIE>
+

dEh~(I2>
V( Il> <21 + 2> <11)

+

IE> <21),

(15)

where I a>, I/I> are two states of a spectator system and it is understood that all terms other than the term in H’ contain the unit operator for this system, I a> , we see that the term in H’ represents a channel—channel coupling between the usual decay channel I Ea> and a new channel I Eli>. Again, on solving we find that, as H’ increases, the width of I la> is first increased from 2xh~,and 2H’_2), then ultimately decreased to zero k~sV The general situation of (as channel—channel coupling may be described by

is not peculiar to the present model, but applies to any channel-channel coupling once effects (a), (b) have been allowed for. A particular situation that has been shown to lead to reduction is that of decay with barrier penetration in the presence of coupling to an oscillatorbath [19]. In terms of the three effects listed above, (ii) is excluded because it is ensured that there is no energy shift. Effect (i) is present, since the coupling to the bath is larger in the channel I E> than in the decaying state I 1> (being proportional to the particle coordinate). Thus differently, and one the can states expect Ii>,a small IE> are Franck—Con“dressed” don factor. Effect (iii) must also be presentfor strong enough coupling to the bath. The calculation reveals a reduction in decay, as one expects qualitatively from effects (i) and (iii). 433

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In summary, we have pointed out the strong conceptual link between several previously unrelated phenomena and analyses in the area of transitions exposed to disturbances. With the aid of simple models, we see that, for sufficiently frequent of strong disturbances, they show a “Zeno effect” whereby transitions are reduced and ultimately prevented altogether, although no observations are involved. Work described in this Letter was undertaken as part of the Underlying Research Programme of the UKAEA. References [lID. HomeandM.A.B. Whitaker,J. Phys. A 19(1986)1847. [2] A. Sudbery, Ann. Phys. (NY) 157 (1984) 512. [3]J. Maddox, Nature 306 (1983) 111.

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[4] A.M. Lane, Phys. Lett. A 99 (1983) 359. [5] L. Allen and J.H. Eberly, Optical resonances and two-level atoms (Wiley, New York, 1975) ch. 6. [6] R.A. Smith, Proc. R. Soc. A 368 (1979) 163. [7] G.J. Milburn, submitted for publication (1988). [8] G. Feinberg and S. Weinberg, Phys. Rev. 123 (1961) 1439. [9] M. Kus and M. Lewenstein, J. Phys. A 19 (1986) 305. [10] A.J. Bray and M.A. Moore, Phys. Rev. Lett. 49 (1982)1545. [11] S. Chakravarty and A.J. Leggett, Phys. Rev. Lett. 52 (1984) [12] KG. Chetyrkin, M.V. Kazarnovsky, V.A. Kuzmin and M.E. Shaposnikov, Phys. Lett. B 99 (1981) 358. [13] P.G.H. Sanders, J. Phys. G 6 (1980) L16l. [14] P.T. Greenland and A.M. Lane, Phys. Lett. A 117 (1986) 181. [15] L.P. Horwitz and E. Katznelson, Phys. Rev. Lett. 50 (1983) 1184. [16] L. Fonda, G.C. Ghirardi and T. Weber, Phys. Lett. B 131 (1983) 309. [111A.M. Lane, Phys. Lett. A 104 (1984) 91. [18] K. Cahill, Phys. Rev. Lett. 51(1983)1600.